A093915 -id:A093915 - cp's OEIS frontend

A093918 a(2k-1)=(2k-1)^2+k, a(2k)=6k^2+k+1: Last term in rows of triangle A093915.

2, 8, 11, 27, 28, 58, 53, 101, 86, 156, 127, 223, 176, 302, 233, 393, 298, 496, 371, 611, 452, 738, 541, 877, 638, 1028, 743, 1191, 856, 1366, 977, 1553, 1106, 1752, 1243, 1963, 1388, 2186, 1541, 2421, 1702, 2668, 1871, 2927, 2048, 3198, 2233, 3481, 2426, 3776

Offset: 1

Amarnath Murthy, Apr 25 2004

Initially defined as "leading diagonal" of the triangle A093915, a(n) is the last term in row n of A093915, i.e. a(n)=A093916(n)+n-1. - M. F. Hasler, Apr 04 2009

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A093915, A093916, A093917.

LinearRecurrence[{0,3,0,-3,0,1},{2,8,11,27,28,58},50] (* Harvey P. Dale, Oct 22 2013 *)

A093918(n)=if(n%2,n^2,6*(n\2)^2)+n\2+1 \\ M. F. Hasler, Apr 04 2009

Equals A093915 o A000217 = A093916 + A023443. - M. F. Hasler, Apr 04 2009

a(n) = (3+(-1)^n+2*n+(5+(-1)^n)*n^2)/4. a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: -x*(x^2+2*x+2)*(x^3-x^2+3*x+1) / ((x-1)^3*(x+1)^3). - Colin Barker, Dec 18 2012

Edited and extended beyond a(6) by M. F. Hasler, Apr 04 2009

A093916 a(2k-1) = (2k-1)^2 + 2 - k, a(2k) = 6k^2 + 2 - k: First column of the triangle A093915.

Original entry on oeis.org

2, 7, 9, 24, 24, 53, 47, 94, 78, 147, 117, 212, 164, 289, 219, 378, 282, 479, 353, 592, 432, 717, 519, 854, 614, 1003, 717, 1164, 828, 1337, 947, 1522, 1074, 1719, 1209, 1928, 1352, 2149, 1503, 2382, 1662, 2627, 1829, 2884, 2004, 3153, 2187, 3434, 2378, 3727, 2577, 4032, 2784, 4349, 2999, 4678, 3222, 5019, 3453, 5372

Offset: 1

Amarnath Murthy, Apr 25 2004

The sequence was initially defined as the first column of the triangle A093915, constructed by trial and error. It is however easy to prove that the sum of the r-th row of A093915, A093917(r), equals twice A006003(r) when r is odd, and three times A006003(r) when r is even. Given the expression of the row sum A093917(r) in terms of the first element a(r), one obtains the explicit formula for a(r). - M. F. Hasler, Apr 04 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A093915, A093917, A093918.

[(n*(5*n-2) + (-1)^n*(n^2+1) + 7)/4: n in [1..70]]; // G. C. Greubel, Dec 30 2021

LinearRecurrence[{0,3,0,-3,0,1},{2,7,9,24,24,53},80] (* Harvey P. Dale, Nov 24 2017 *)

A093916(n)=((n^2+1)*(3-n%2)-n+1)/2
/* or the "experimental" version, trying out all allowed values */
A093916(n)={ local( s=(n^3+n)/2, d=(n^2-n)/2, k=ceil((2*s-d)/n)); while( (n*k+d)%s, k++ ); k } \\ M. F. Hasler, Apr 04 2009

[(5*n^2 -2*n +7 +(-1)^n*(n^2 +1))/4 for n in (1..70)] # G. C. Greubel, Dec 30 2021

a(n) = ((n^2+1)*b(n) - n + 1)/2 where b(n) = 3 - (n mod 2) = 2 if n odd, = 3 if n even. - M. F. Hasler, Apr 04 2009
From Colin Barker, May 01 2012: (Start)
a(n) = (n*(5*n-2) + (n^2+1)*(-1)^n + 7)/4.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
G.f.: x*(2+7*x+3*x^2+3*x^3+3*x^4+2*x^5)/((1-x)^3*(1+x)^3). (End)
E.g.f.: (1/4)*( (7 +3*x +5*x^2)*exp(x) - 8 + (1 -x +x^2)*exp(-x) ). - G. C. Greubel, Dec 30 2021

Edited and extended by M. F. Hasler, Apr 04 2009

A093917 a(n) = n^3+n for odd n, (n^3+n)*3/2 for even n: Row sums of A093915.

Original entry on oeis.org

2, 15, 30, 102, 130, 333, 350, 780, 738, 1515, 1342, 2610, 2210, 4137, 3390, 6168, 4930, 8775, 6878, 12030, 9282, 16005, 12190, 20772, 15650, 26403, 19710, 32970, 24418, 40545, 29822, 49200, 35970, 59007, 42910, 70038, 50690, 82365, 59358

Offset: 1

Amarnath Murthy, Apr 25 2004

Initially defined as sum of the n-th row of the triangle A093915, constructed by trial and error. Namely, this row should contain n consecutive integers [x,x+1,...,x+n-1], listed in A093915, and have its sum a(n) = n*x+n(n-1)/2 equal to the least possible strict (>1) multiple of the sum of the indices of these elements in A093915, which equals A006003(n) = (n^3+n)/2. For odd n, a(n) = 2 A006003(n) is obtained for x = A093916(n). For even n, the sum a(n) cannot equal 2 A006003(n), but it does equal 3 A006003(n) for x = A093916(n). Hence this simple explicit definition of a(n). - M. F. Hasler, Apr 04 2009

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A093915, A093916, A093918.

a(n) = n*A093916(n)+n(n-1)/2. - M. F. Hasler, Apr 04 2009
a(2n-1) = 2*(2n-1)*(2n^2 -2n +1), a(2n) = 3*n*(4n^2 +1).
G.f.: x*(2+15*x+22*x^2+42*x^3+22*x^4+15*x^5+2*x^6) / ( (x-1)^4*(1+x)^4 ). - R. J. Mathar, Mar 21 2016

More terms from Jorge Coveiro, Jul 25 2006

Edited by M. F. Hasler, Apr 04 2009

cp's OEIS Frontend

A093918 a(2k-1)=(2k-1)^2+k, a(2k)=6k^2+k+1: Last term in rows of triangle A093915.

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A093916 a(2k-1) = (2k-1)^2 + 2 - k, a(2k) = 6k^2 + 2 - k: First column of the triangle A093915.

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A093917 a(n) = n^3+n for odd n, (n^3+n)*3/2 for even n: Row sums of A093915.

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cp's OEIS Frontend

A093918 a(2k-1)=(2k-1)^2+k, a(2k)=6k^2+k+1: Last term in rows of triangle A093915.

Views

Author

Keywords

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Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A093916 a(2*k-1) = (2*k-1)^2 + 2 - k, a(2*k) = 6*k^2 + 2 - k: First column of the triangle A093915.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A093917 a(n) = n^3+n for odd n, (n^3+n)*3/2 for even n: Row sums of A093915.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A093916 a(2k-1) = (2k-1)^2 + 2 - k, a(2k) = 6k^2 + 2 - k: First column of the triangle A093915.