A076389 -id:A076389 - cp's OEIS frontend

A076465 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly n ways.

1, 571, 12938, 115270, 630755, 2543401, 8307796, 23249388, 57792165, 130790935, 274285726, 540036146, 1008233863, 1798831685, 3085968040, 5116005976, 8229746121, 12889413363, 19711057330, 29503047070, 43311380651, 62472570721, 88674907388, 124028940100

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A076389, A076459-A076464.

[n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1)/6: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/6*n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1),n=1..25);

CoefficientList[Series[(1 + 562 x + 7835 x^2 + 19300 x^3 + 11255 x^4 + 1354 x^5 + 13 x^6)/(1 - x)^9, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,571,12938,115270,630755,2543401,8307796,23249388,57792165},30] (* Harvey P. Dale, Sep 05 2015 *)

a(n) = n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1)/6.
G.f.: x*(1+562*x+7835*x^2+19300*x^3+11255*x^4+1354*x^5+13*x^6)/(1-x)^9.
a(1)=1, a(2)=571, a(3)=12938, a(4)=115270, a(5)=630755, a(6)=2543401, a(7)=8307796, a(8)=23249388, a(9)=57792165, a(n)=9*a(n-1)- 36*a(n-2)+ 84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Sep 05 2015

More terms from Vincenzo Librandi, Dec 30 2013

A076460 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly one way.

Original entry on oeis.org

1, 103, 1130, 6070, 22355, 64981, 160468, 351660, 703365, 1308835, 2297086, 3841058, 6166615, 9562385, 14390440, 21097816, 30228873, 42438495, 58506130, 79350670, 106046171, 139838413, 182162300, 234660100, 299200525, 377898651, 473136678, 587585530, 724227295

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A076389, A076461-A076465.

[n*(n+1)*(7*n^4+2*n^3-6*n^2-n+1)/6: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/6*n*(n+1)*(7*n^4+2*n^3-6*n^2-n+1),n=1..30);

CoefficientList[Series[(1 + 96 x + 430 x^2 + 288 x^3 + 25 x^4)/(1 - x)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,103,1130,6070,22355,64981,160468},30] (* Harvey P. Dale, Jul 04 2025 *)

a(n) = n*(n+1)*(7*n^4+2*n^3-6*n^2-n+1)/6.

G.f.: x*(1+96*x+430*x^2+288*x^3+25*x^4)/(1-x)^7.

More terms from Vincenzo Librandi, Dec 30 2013

A076461 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly two ways.

Original entry on oeis.org

13, 571, 5306, 26470, 93455, 264313, 640276, 1383276, 2736465, 5047735, 8796238, 14621906, 23357971, 36066485, 54076840, 79027288, 112909461, 158115891, 217490530, 294382270, 392701463, 516979441, 672431036, 865020100, 1101528025, 1389625263, 1737945846, 2156164906

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A076389, A076460-A076465.

[n*(n+1)*(25*n^4+26*n^3-6*n^2-7*n+1)/6: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/6*n*(n+1)*(25*n^4+26*n^3-6*n^2-7*n+1),n=1..30);

CoefficientList[Series[(13 + 480 x + 1582 x^2 + 864 x^3 + 61 x^4)/(1 - x)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(25*n^4+26*n^3-6*n^2-7*n+1)/6.

G.f.: x*(13+480*x+1582*x^2+864*x^3+61*x^4)/(1-x)^7.

More terms from Vincenzo Librandi, Dec 30 2013

A076464 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly five ways.

Original entry on oeis.org

145, 4567, 38570, 183670, 630755, 1751365, 4187092, 8957100, 17583765, 32236435, 55893310, 92521442, 147274855, 226710785, 339024040, 494299480, 704782617, 985168335, 1352907730, 1828533070, 2436000875, 3203053117, 4161596540, 5348100100, 6804010525

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A076389, A076460-A076465.

[n*(n+1)*(151*n^4+242*n^3+66*n^2-25*n+1)/6: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/6*n*(n+1)*(151*n^4+242*n^3+66*n^2-25*n+1),n=1..30);

CoefficientList[Series[(145 + 3552 x + 9646 x^2 + 4512 x^3 + 265 x^4)/(1 - x)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(151*n^4+242*n^3+66*n^2-25*n+1)/6.

G.f.: x*(145+3552*x+9646*x^2+4512*x^3+265*x^4)/(1-x)^7.

More terms from Vincenzo Librandi, Dec 30 2013

A076430 Sum of squares of numbers that cannot be written as tp(n) + up(n+1) for nonnegative integers t,u, where p(n) is the n-th prime.

Original entry on oeis.org

1, 70, 1610, 22715, 170170, 675376, 2224824, 5696295, 21057190, 52798270, 111385170, 261955560, 414410780, 628310711, 1187225546, 2369546790, 3631067090, 5339542890, 8459444455, 10969197540, 15154637940, 22349763995

Offset: 1

Floor van Lamoen, Oct 10 2002

nonn

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Cf. A076389, A076429.

seq(ithprime(n)*ithprime(n+1)*(ithprime(n)-1)*(ithprime(n+1)-1)*(ithprime(n)*ithprime(n+1)-ithprime(n)-ithprime(n+1))/12,n=1..30);

a(n) = p(n) * p(n+1) * (p(n)-1) * (p(n+1)-1) * (p(n)*p(n+1)-p(n)-p(n+1)) / 12.

A076462 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly three ways.

Original entry on oeis.org

41, 1471, 12938, 62870, 218555, 611821, 1471316, 3161388, 6227565, 11448635, 19895326, 32995586, 52606463, 81092585, 121411240, 177204056, 252895281, 353796663, 486218930, 657589870, 876579011, 1153228901, 1499092988, 1927380100, 2453105525, 3093248691

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A076389, A076460-A076465.

[n*(n+1)*(55*n^4+74*n^3+6*n^2-13*n+1)/6: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/6*n*(n+1)*(55*n^4+74*n^3+6*n^2-13*n+1),n=1..30);

CoefficientList[Series[(41 + 1184 x + 3502 x^2 + 1760 x^3 + 113 x^4)/(1 - x)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(55*n^4+74*n^3+6*n^2-13*n+1)/6.

G.f.: x*(41+1184*x+3502*x^2+1760*x^3+113*x^4)/(1-x)^7.

A076463 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly four ways.

Original entry on oeis.org

85, 2803, 24026, 115270, 397655, 1107505, 2653588, 5685996, 11176665, 20511535, 35594350, 58962098, 93912091, 144640685, 216393640, 315628120, 450186333, 629480811, 864691330, 1168973470, 1557678815, 2048586793, 2662148156, 3421740100, 4353933025

Offset: 1

Floor van Lamoen, Oct 13 2002

Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A076389, A076460-A076465.

[n*(n+1)*(97*n^4+146*n^3+30*n^2-19*n+1)/6: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

seq(1/6*n*(n+1)*(97*n^4+146*n^3+30*n^2-19*n+1),n=1..30);

CoefficientList[Series[(85 + 2208 x + 6190 x^2 + 2976 x^3 + 181 x^4)/(1 - x)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)

a(n) = n*(n+1)*(97*n^4+146*n^3+30*n^2-19*n+1)/6.

G.f.: x*(85+2208*x+6190*x^2+2976*x^3+181*x^4)/(1-x)^7.

cp's OEIS Frontend

A076465 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly n ways.

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A076460 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly one way.

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A076461 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly two ways.

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A076464 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly five ways.

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A076430 Sum of squares of numbers that cannot be written as t*p(n) + u*p(n+1) for nonnegative integers t,u, where p(n) is the n-th prime.

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A076462 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly three ways.

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Magma

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A076463 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly four ways.

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A076465 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly n ways.

A076460 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly one way.

A076461 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly two ways.

A076464 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly five ways.

A076430 Sum of squares of numbers that cannot be written as tp(n) + up(n+1) for nonnegative integers t,u, where p(n) is the n-th prime.

A076462 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly three ways.

A076463 Sum of squares of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly four ways.