A076455 -id:A076455 - cp's OEIS frontend

A076454 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly one way.

1, 21, 102, 310, 735, 1491, 2716, 4572, 7245, 10945, 15906, 22386, 30667, 41055, 53880, 69496, 88281, 110637, 136990, 167790, 203511, 244651, 291732, 345300, 405925, 474201, 550746, 636202, 731235, 836535, 952816, 1080816, 1221297, 1375045, 1542870, 1725606, 1924111

Offset: 1

Floor van Lamoen, Oct 13 2002

This sequence is related to A007585 by a(n) = n*A007585(n) - Sum_{i=0..n-1} A007585(i). - Vincenzo Librandi, Aug 08 2010
In fact, this is the case d=4 in the identity n*(n*(n+1)*(2*d*n-2*d+3)/6) - Sum_{k=0..n-1} k*(k+1)*(2*d*k-2*d+3)/6 = n*(n+1)*(3*d*n^2-d*n+4*n-2*d+2)/12. - Bruno Berselli, Mar 01 2012
Bisection of A233329 (up to an offset). - L. Edson Jeffery, Jan 23 2014

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
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002417, A007585, A076455-A076459, A233329.

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

Maple
```
seq(1/2*n*(n+1)*(2*n^2-1),n=1..40);
```

CoefficientList[Series[(1 + 16 x + 7 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,21,102,310,735},40] (* Harvey P. Dale, Jun 30 2023 *)

a(n) = n*(n+1)*(2*n^2-1)/2.
G.f.: x*(1+16*x+7*x^2)/(1-x)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), n>=6, with a(1)=1, a(2)=21, a(3)=102, a(4)=310, a(5)=735. - L. Edson Jeffery, Dec 30 2013

Comments rewritten from Bruno Berselli, Mar 01 2012

More terms from Vincenzo Librandi, Dec 30 2013

A304659 a(n) = n(n + 1)(16*n - 1)/6.

Original entry on oeis.org

0, 5, 31, 94, 210, 395, 665, 1036, 1524, 2145, 2915, 3850, 4966, 6279, 7805, 9560, 11560, 13821, 16359, 19190, 22330, 25795, 29601, 33764, 38300, 43225, 48555, 54306, 60494, 67135, 74245, 81840, 89936, 98549, 107695, 117390, 127650, 138491, 149929, 161980, 174660, 187985

Offset: 0

Bruno Berselli, May 22 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A007742, A076455, A139273 (first differences).

First lower diagonal of the rectangular array in A213835.

[n*(n+1)*(16*n-1)/6: n in [0..41]]; // Vincenzo Librandi, May 23 2018

Table[n (n + 1) (16 n - 1)/6, {n, 0, 50}]

concat(0, Vec(x*(5 + 11*x) / (1 - x)^4 + O(x^40))) \\ Colin Barker, May 25 2018

O.g.f.: x*(5 + 11*x)/(1 - x)^4.
E.g.f.: x*(30 + 63*x + 16*x^2)*exp(x)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) + a(-n) = A033429(n).
a(n) = n*A007742(n) - Sum_{k = 0..n-1} A007742(k) for n > 0.
Also, this sequence is related to A076455 by the same type of recurrence:
A076455(n) = n*a(n) - Sum_{k = 0..n-1} a(k) for n > 0.

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A076454 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly one way.

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A304659 a(n) = n(n + 1)(16*n - 1)/6.

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cp's OEIS Frontend

A076454 Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly one way.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A304659 a(n) = n*(n + 1)*(16*n - 1)/6.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A076454 Sum of numbers that can be written as tn + u(n+1) for nonnegative integers t,u in exactly one way.

A304659 a(n) = n(n + 1)(16*n - 1)/6.