A255178 Second differences of eighth powers (A001016).

1, 254, 6050, 52670, 266114, 963902, 2796194, 6927230, 15257090, 30683774, 57405602, 101263934, 170126210, 274309310, 427043234, 644975102, 948713474, 1363412990, 1919399330, 2652834494, 3606422402, 4830154814, 6382097570, 8329217150, 10748247554

Offset: 0

Author

Luciano Ancora, Feb 21 2015

Keywords

• nonn
• easy

Examples

			Second differences:  1, 254, 6050, 52670, 266114, ... (this sequence)
First differences:   1, 255, 6305, 58975, 325089, ... (A022524)
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The eighth powers:   1, 256, 6561, 65536, 390625, ... (A001016)
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First partial sums:  1, 257, 6818, 72354, 462979, ... (A000542)
Second partial sums: 1, 258, 7076, 79430, 542409, ... (A253636)
Third partial sums:  1, 259, 7335, 86765, 629174, ... (A254642)
Fourth partial sums: 1, 260, 7595, 94360, 723534, ... (A254647)
		

Links

• Luciano Ancora, Table of n, a(n) for n = 0..1000
• Luciano Ancora, Sums of powers of positive integers and their recurrence relations, section 0.5.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000542, A001016, A022524, A253636, A254642, A254647, A255177.

Programs

Magma

[n eq 0 select 1 else 2*(28*n^6+70*n^4+28*n^2+1): n in [0..30]]; // Vincenzo Librandi, Mar 12 2015

Mathematica

Join[{1}, Table[2 (28 n^6 + 70 n^4 + 28 n^2 + 1), {n, 1, 30}]]
Join[{1},Differences[Range[0,30]^8,2]] (* Harvey P. Dale, Aug 26 2024 *)

Formula

G.f.: (1 + x)*(1 + 246*x + 4047*x^2 + 11572*x^3 + 4047*x^4 + 246*x^5 + x^6)/(1 - x)^7.

a(n) = 2*(28*n^6 + 70*n^4 + 28*n^2 + 1) for n>0, a(0)=1.

Extensions

Edited by Bruno Berselli, Mar 19 2015